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Inner product spaces of integrands associated to subfractional Brownian motion. (English) Zbl 1283.60082

Summary: We characterize the domain of the Wiener integral with respect to a subfractional Brownian motion \(\{S_H(t)\}_{t\geq0}\), \(H\in(0,1)\), \(H\neq\frac{1}{2}\). The domain is a Hilbert space which contains the class of elementary functions as a dense subset. If \(0<H<\frac{1}{2}\), any element of the domain is a function and if \(\frac{1}{2}<H<1\), the domain is a space of distributions. The RKHS of \(S_H\) is also determined.

MSC:

60H05 Stochastic integrals
60B11 Probability theory on linear topological spaces
60G22 Fractional processes, including fractional Brownian motion
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[1] Bojdecki, T.; Gorostiza, L.; Talarczyk, A., Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett., 69, 405-419 (2004) · Zbl 1076.60027
[2] Bojdecki, T.; Gorostiza, L.; Talarczyk, A., Fractional Brownian density process and its self-intersection local time of order \(k\), J. Theoret. Probab., 69, 5, 717-739 (2004) · Zbl 1074.60047
[3] Bojdecki, T.; Gorostiza, L.; Talarczyk, A., Limit theorems for occupation time fluctuations of branching systems 1: Long-range dependence, Stochastic. Process. Appl., 116, 1-18 (2006) · Zbl 1082.60024
[4] Gel’fand, I. M.; Shilov, G. E., (Properties and Operations. Properties and Operations, Generalized Functions, vol. 1 (1964), Academic Press: Academic Press New York) · Zbl 0115.33101
[5] Grenander, U., Abstract Inference (1981), John Wiley & Sons: John Wiley & Sons New York · Zbl 0673.62088
[6] Jolis, M., 2006. The Wiener integral with respect to second order processes with stationary increments. Preprint; Jolis, M., 2006. The Wiener integral with respect to second order processes with stationary increments. Preprint · Zbl 1188.60027
[7] Jolis, M., On the Wiener integral with respect to the fractional Brownian motion on an interval, J. Math. Anal. Appl., 330, 22, 1115-1127 (2007) · Zbl 1185.60057
[8] Kolmogorov, A. N., Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C.R. (Doklady) Acad. URSS (N.S.), 26, 115-118 (1940) · Zbl 0022.36001
[9] Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437 (1968) · Zbl 0179.47801
[10] Neveu, J., Processus Aléatoires Gausienes (1968), Les Presses de L’Université de Montréal · Zbl 0192.54701
[11] Pipiras, V.; Taqqu, M. S., Integration questions related to fractional Brownian motion, Probab. Theory Related Fields, 118, 251-291 (2000) · Zbl 0970.60058
[12] Samko, S,G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (1993), Gordon and Breach Science · Zbl 0818.26003
[13] Tudor, C., Some properties of the sub-fractional Brownian motion, Stochastics, 79, 5, 431-448 (2007) · Zbl 1124.60038
[14] Weinert, H. L., Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing (1982), Hutchinson Ross: Hutchinson Ross Stroudsburg, PA
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